1 33 honeycomb

133 honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3,3^3,3}
Coxeter symbol	133
Coxeter-Dynkin diagram	or
7-face type	132
6-face types	122 131
5-face types	121 {3^4}
4-face type	111 {3^3}
Cell type	101
Face type	{3}
Cell figure	Square
Face figure	Triangular duoprism
Edge figure	Tetrahedral duoprism
Vertex figure	Trirectified 7-simplex
Coxeter group	${\tilde {E}}_{7}$, [[3,3<sup>3,3</sup>]]
Properties	vertex-transitive, facet-transitive

In 7-dimensional geometry, 1₃₃ is a uniform honeycomb, also given by Schläfli symbol {3,3^3,3}, and is composed of 1₃₂ facets.

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Removing a node on the end of one of the 3-length branch leaves the 1₃₂, its only facet type.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 0₃₃.

The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.

Kissing number

Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.

Geometric folding

The ${\tilde {E}}_{7}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

${\tilde {E}}_{7}$	${\tilde {F}}_{4}$
{3,3^3,3}	{3,3,4,3}

E₇^* lattice

${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144.^[1] Both ${\tilde {E}}_{7}$ and ${\tilde {A}}_{7}$ can be seen as affine extension from $A_{7}$ from different nodes:

The E₇^* lattice (also called E₇²)^[2] has double the symmetry, represented by [[3,3^3,3]]. The Voronoi cell of the E₇^* lattice is the 1₃₂ polytope, and voronoi tessellation the 1₃₃ honeycomb.^[3] The E₇^* lattice is constructed by 2 copies of the E₇ lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A₇^* lattices, also called A₇⁴:

∪ = ∪ ∪ ∪ = dual of .

Related polytopes and honeycombs

The 1₃₃ is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde {E}}_{7}$=E7^+	${\bar {T}}_{8}$=E7^++
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[3^1,3,1]	[3^2,3,1]	[[3<sup>3,3,1</sup>]]	[3^4,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	13,-1	130	131	132	133	134

Rectified 1_33 honeycomb

The rectified 1₃₃ or 0₃₃₁, Coxeter diagram has facets and , and vertex figure .

See also

• 8-polytope
• 3₃₁ honeycomb

Notes

1. N.W. Johnson: Geometries and Transformations, (2015) Chapter 12: Euclidean symmetry groups, p 177
2. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
3. The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin

References

• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
Uniform 5-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ6	hδ6	qδ6
Uniform 7-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
Uniform 8-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
Uniform 9-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
Uniform 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21